2016美赛特等奖原版论文集Jiaotong University China

1、For office use only T1 _ T2 _ T3 _ T4 _ Team Control Number 46830 Problem Chosen A For office use only F1 _ F2 _ F3 _ F4 _ 2016 MCM/ICM Summary Sheet Keep It Warm Summary Do you usually get annoyed with the cooling of water during a relaxing bath? In this paper, several models are proposed to solve

2、the problem. Our efforts in building these models followed the path below. Firstly, a universal thermodynamic model(UTM) of bath water is established by constructing two initial sub-equations named the Heat Differential Equation(HDE) and the Temperature Distribution(TD) respectively. Based on Fourie

3、rs law, the Stefan-Boltzmann law and the first law of thermodynamics, HDE is established, which also takes heat conduction, heat radiation and evaporation of water into account. Moreover, we get the numerical solution of TD by using the PDE tool of MATLAB, and fit its temperature distribution surfac

4、e by the plane equation, whose goodness of fit is 0.931. The fitting equation is regarded as the approximate analytical solution. Secondly, taking convection caused by human movement and comfort of water temperature into account, we developed an optimization control model for the bath strategy (OCM)

5、 based on UTM, which aims at the smallest water injection, the best temperature stability ii. Injecting hot water when temperature declines to min1 T and stopping injecting when temperature increase to min2 T. In strategy I, the optimal conditions are that the flow is 1 0.007L s and the temperature

6、of water injection is 311.1K. The optimization proves efficient after extensive discussion; In strategy II, the upper and lower critical temperature and the temperature of water injection are 311.7K，311.7K ii. The natural cooling process of water; iii. The process of adding hot water. Consider the i

7、nfluences of the motions of people in tub, and analyze the results of this influences. We need to deal with the obstacle caused by the convection of water and the accompanying heat exchange. Find out the best control strategy to keep the temperature and save water as possible. The strategy depends o

8、n the parameters of tub, the degree of human movement, and the parameters of the human body. Discuss the difference between water and bubble bath water. We shall clarify how the bubble bath water influences the model and the strategy we have given. We need to provide a one-page non-technical explana

9、tion for users of the bathtub that describes our strategy. It shall explain why it is so difficult to get an evenly maintained temperature throughout the bath water and the advice according to our model. 1.2 Our Approach We analyzed the problems above and consulted lots of literature, and then come

10、up with the following approaches: We analyzed the problems above and consulted lots of literature, and then come up with the following approaches: Team #46830 Page 2 of 45 We build a general thermodynamic model for water bath. This model is based on the Fourier law, the Newton cooling theorem, Stern

11、-boltzmanns law, the heat conduction formula, and the first law of thermodynamics. And it includes: I.Heat exchange between hot water and bathtub II. Heat exchange between hot water and air, III.Evaporation heat absorption of hot water, IV.Heat absorption and heat release of human body, V.Heat radia

12、tion of water, VI. Heat of added hot water and heat carried away by the overflow. We build a distribution model for water in bathtub. This model is mainly based on the heat conduction equation and the Neumann boundary condition. Unfortunately, the model can only be solved by Matlab, and cant get acc

13、urate analytic function expression. So we build an approximate analytical model for water temperature distribution. We build a forced convection model for the water in the bathtub. The motion of human body in the water will be the force of water. And the convection of water accelerates the heat exch

14、ange of water. So we quantify human motion, then build a model for forced convection heat exchanging. This model describes the redistribution of the water temperature under the influence of human motion. We build an optimal control strategy model based on thermodynamics, water temperature distributi

15、on and forced convection. The model is mainly based on nonlinear multivariable optimal control, use it to get the optimal control strategy. We will provide two control strategies based on different aspects under this model. We obtain the thermodynamic model for bubble bath water by improving general

16、 thermodynamic model. Based on this model, we discussed the effects of bubble bath water on the distribution and variation of water temperature, then analyzes the influence of the bubble bath water to the optimal control strategy. At the same time, we analyzed the advantages of bubble bath water. We

17、 use the finite difference method to solve the partial differential equations and the piecewise differential equations. 2 Assumptions with Justifications We make the following assumptions about the whole process in this paper to obtain a better model result. Team #46830 Page 3 of 45 2.1 About the Ba

18、throom and Water 1. Assume that the air temperature and humidity in the bathroom is constant, not due to the Water vapor from man and bathtub. The temperature and humidity distribution of air is affected by many factors, this problem is very difficult. The other reason is that the discussion of air

19、temperature and humidity not mean much. 2. Without taking into account the wave generated by the motion of human and the adding of water, as well as the overflow caused by waves. The reason is that the water wave will be reflected back by the Inner wall of bathtub and leads multiple interference, it

20、 is very difficult to study this process. In addition, this mechanical process will not have a significant impact on the thermodynamic process of the bathtub-hot water-human system. 3. Regardless of the time between the hot water comes out of the tap and enters the bathtub, which means that ignoring

21、 heat dissipation of water in the air. The reason is that the setting position of the tab should be as close as possible to the surface of the full water in the bathtub. That means the time for water to stay in the air is so short that the amount of heat dissipation in the air is negligible. 4. Natu

22、ral convection due to temperature gradient is negligible. Consider only the forced convection caused by the movement of the human. Because the temperature gradient of the bath water is not large enough to consider the natural convection of water caused by it. 5. The convection of water inside the ba

23、th tub does not affect the current water temperature. The reason is that heat exchange caused by convective occurs only between the parts of the water in the bathtub, this process does not get heat exchange from the outside of the system, in other words, only water temperature distribution changed.

24、6. The physical parameters such as the thermal conductivity, density and the bubble thickness are not changed with the addition of hot water and the loss of overflow water. Although the concentration of bubble water is bound to change with the addition and loss of water, the influence of bubble wate

25、r concentration on other physical parameters is complicated and not clear, and the change of concentration is quite small for some parameters. Therefore, ignore the bubble water physical properties change over time 2.2 About the Human Body 7. Heat conductivity of human body is the constant value, it

26、 do not change with human posture and body temperature. Because the heat absorption and release of the Team #46830 Page 4 of 45 human body is a complex process of biochemical reaction and physical process. Some factors like the metabolic heat generation, human sweat evaporation and the work done by

27、exercise need to be considered. Since the thermodynamic process in the human body is too complicated and difficult to be analyze, considered the human body as a constant temperature heat source (maybe its endothermic), its thermal conductivity is a constant value. 8. Each part of the body surface te

28、mperature is a constant value. Because the distribution of body temperature is difficult to parameterized, and the body temperature of each part is slightly different(around 0.6). Consider that all over the body surface, the temperatures are the same constant. 9. Ignore the work of the forced convec

29、tion generated by human motion. As the fluid, water can only work rely on its viscous force, and the viscous force is too small to be taken into account. 10. Considered the human motion as a series of discrete motion with the same amplitude and frequency. Although the motions can have a huge impact

30、on the environment, such as the spatial distribution of water temperature, the movement of people is very irregular. So we use a simple physical quantity to describe the intensity and speed of human motion. For the magnitude of the motions, we will discuss it in detail in the 4.2 chapter. 3 Notation

31、s All the variables and constants used in this paper are listed in Table1 and Table 2. Table 1 Symbol TableConstants Symbol Definition Value 0 T Air temperature 298K P Saturated vapor pressure 3 2.4 10 Pa Density of water 33 10 kg m Molar mass of water 31 18 10 kg mol R Universal gas constant 11 8.3

32、1J molK c Specific heat capacity of water 311 4.2 10 J kgK Surface radiation coefficient of water 0.97 Stefan-Boltzmann constant 824 5.67 10 W mK Team #46830 Page 5 of 45 Table 2 Symbol TableVariables Symbol Definition Units Q Internal energy of the water in the bathtub J ( )T t Average temperature

33、of water at time t K exchange Q Heat of heat exchange J evaporation Q Heat of water evaporation J body Q Heat absorbed by the body J radiation Q Heat of radiation J ( , , , )T x y z t Distribution of temperature in time and space K ( )u T Water evaporation rate 1 m s k Water flow 31 ms 0 T The outle

34、t water temperature K k S The movement distance of the kth m k V The volume distance of the kth 3 m F Source of forced convection 4 m b k Best water flow 31 ms b T The best outlet water temperature K 4 The Model We establish an optimal control strategy model based on our normal thermodynamic model,

35、forced convection model of water temperature and body temperature comfort model. Here are the relationship among the three models and equations. Initial Sub equation Secondary Sub model Final model Heat differential equationTemperature distribution Universal thermodynamic model Forced convection mod

36、el Body temperature comfort model Optimal control model of the bath strategy Team #46830 Page 6 of 45 Figure 1: The relationship among the models and equations According to our model, we propose two kinds of temperature control strategies and analyze their both advantages and shortages. In addition,

37、 the shape and volume of the tub, the temperature, flow and convection of tap water, the motions made by the person and some other parameters are extensively discussed in this section. The establishment and solution of our models are as follow. 4.1 Normal Thermodynamic Model of Water Temperature In

38、this section, we set up a normal thermodynamic model of water temperature in space and time, which is based on Fouriers law, Newtons law of cooling, the Stefan-Boltzmann law, the heat conduction equation and the first law of thermodynamics. We use Newtons law of cooling, the Stefan-Boltzmann law and

39、 the first law of thermodynamics to solve the problem of the distribution of temperature with time. And we use heat conduction equation to solve the problem of the distribution of temperature with space. In our model, the heat transfer and the heat radiation are taken into account while the convecti

40、on is ignored. 4.1.1 The differential equation of water heat in the tub Considering the origin and destination of the infinitesimal of water heat in the tub, we get the differential equation as follows exchangeevaporationbodyradiationwaterwater dQdQdQdQdQdQdQ (1) Here are the derivation and explanat

41、ion of the equation by analyzing the heat of each part. First of all, assuming the tub as a cube with length a, width b and height H. Then we set a three-dimensional Cartesian coordinate system originated from the bottom center of the tub and paralleled to its height, length and width. As we all kno

42、w, the drain and overflow valve of the tub are often designed in the midpoint of the two short sides. Thus, the overflow valve is located in 0, 2 a H and the drain is in 0, 2 a H . Team #46830 Page 7 of 45 Figure 2: the three-dimensional Cartesian coordinate system of the tub To establish the model

43、of water temperature over time, we firstly consider that the volume of water is changing over time. When the tub isnt full, the volume of water is proportional to the time growth. Because we usually open the valve to the maximum when we are going to take a shower, the growth rate of the volume of wa

44、ter over time is the biggest flow of the valve. Then, when the volume is proper, people enter the tub and from then on, the volume can be regarded as constant approximately. Here comes the equation. 0 max max 0 0 max 0 V ktt k V V Vt k Where, 0body VabHV, it means that the largest volume of water wh

45、en the person is in the tub without overflow. max k is the largest flow of the valve. By the definition of internal energy, we get the equation of the energy of water in the tub. ( )Qc T tdV (2) Where, ( )T t is the average water temperature on space, which can be defined by ( , , , ) ( ) T x y z td

46、V T t dV (3) Then, we analyze different part of the heat exchange. The heat exchange is composed of heat transfer, heat radiation, convection, the inflow of hot water and the outflow of cold water. Here we discussed them except convection which is discussed in 4.2. First, we analyze the heat loss du

47、e to heat exchange5. The loss is caused by the Team #46830 Page 8 of 45 contact of water with the inner wall of the tub and air. According to the Fouriers law, the heat transfer power of per unit area is proportional to the temperature difference and to the thickness of the heat transfer medium. If

48、we suppose that the thickness of the tub wall is constant and its material is homogeneous, the heat transfer power of per unit area is proportional to the temperature difference only. Thus we get the equation of the whole heat emission in unit time. 00 ( , , , ) ( , , , ) exchange D dQh T x y z tT d

49、Ah T x y z tT dA dt (4) Where, h is the heat transfer coefficient of water when the medium is the tub wall. h is the heat transfer coefficient of water to air. Secondly, we analyze the heat loss due to water evaporation. According to the surface molecular evolution rule and Daltons law of evaporatio

50、n, the evaporation rate per unit area is related to the liquid temperature, surface wind speed and humidity. Because of assumption 1, humidity is constant and the evaporation rate per unit area is related to the liquid temperature. Here comes the following equation. 1 2 0 1 2 evaporationgas D dQu Tq

51、dAdt u Tw PRT (5) Where, u T is the rate of water evaporation per unit time and area; T is the air temperature of the air to water surface; P is the saturated vapor pressure of water at current temperature; w is the relative humidity of air in the bathroom, which is constant according to assumption

52、1; 33 10 kg m , which is the density of water ; 31 18 10 kg mol , which is the mole mass of water; 11 8.31RJ molk , which is the universal gas constant; Third, we analyze the heat loss due to peoples body 7. To be specific, it describes the heat exchange of peoples body surface and water. We suppose

53、 that the body is fully immersed in water, so the contact area between body and water is the superficial area of the body. Because the heat exchange between the human body and the water is the heat transfer through the water flow, according to Newton cooling theorem, we get the heat infinitesimal eq

54、uation of bodys absorption. Team #46830 Page 9 of 45 ( ) bodybodybodybody dQST tThdt (6) Where, body h is the heat transfer rate of human body and water. Because the similarity of the macroscopic physical properties between human body and water, body h can be substituted with heat transfer rate of w

55、ater and water. ( )T t is the average water temperature in the tub. The reason to set this parameter is because the surface of human body is irregular, the temperature distribution between the contact surfaces is harder to formulate. Then we regard the average water temperature in the tub as that of

56、 the contact surfaces. Fourth, we analyze the heat loss due to heat radiation 10. To be specific, it describes that water dissipates heat in the form of electromagnetic radiation to the surrounding and the loss can be calculated by Trevor Boltzmanns law. Unfortunately, the law is set to describe the

57、 heat radiation of ideal black-body, while water isnt totally suit. Thus we modify the law with a rate of heat radiation , and it is recorded that the rate of water is 0.97. we get the heat infinitesimal equation of the loss. 4 , , , radiation dQTx y z t dA dt (7) Fifth, we analyze the heat increase

58、 due to the constant intake of hot water. According to the definition of the internal energy of water, we get the heat infinitesimal equation of the increase. 00water dQc T dVc T kdt (8) Sixth, we analyze the heat loss due to the overflow water. We assume the overflow water temperature as the temperature of the water near the outfall and get the h